The graph of a sinusoidal function has a minimum point at $(0,-10)$ and then has a maximum point at $(2,-4)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
Answer: The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline passes exactly between the minimum value $-10$ and the maximum value $-4$, so the midline equation is $y={-7}$. The extremum points are $3$ units above or below the midline, so the amplitude is ${3}$. The maximum point is $2$ units to the right of the minimum point, so the period is $2\cdot 2={4}$. [Why did we multiply by 2?] Determining the type of function to use Since the graph has an extremum point at $x=0$, we should use the cosine function and not the sine function. This means there's no horizontal shift, so the function is of the form $a\cos(bx)+d$. [How do we know that?] Determining the parameters in $a\cos(bx)+d$ Since the extremum point at $x=0$ is a minimum point, we know that $a<0$. [How do we know that?] The amplitude is ${3}$, so $|a|={3}$. Since $a<0$, we can conclude that $a=-3$. The midline is $y={-7}$, so $d=-7$. The period is ${4}$, so $b=\dfrac{2\pi}{{4}}=\dfrac{\pi}{2}$. The answer $f(x)=-3\cos\left(\dfrac{\pi}{2}x\right)-7$